${\stackrel{\sim }{T}}_i=q^{–1/2}{T}_i$ so that ${\stackrel{\sim }{T}}_i^2=(q^{1/2}–q^{–1/2}){\stackrel{\sim }{T}}_i+1$.

Define a new basis $\{x_v\mid v\in \stackrel{\sim }{W}\}$ by $x_g=T_g$ for $\ell \left(g\right)=0$ and

$${\stackrel{\sim }{T}}_i{x}^{{\mu }^{\vee }}=x^{s_i{\mu }^{\vee }}{\stackrel{\sim }{T}}_i+(q^{1/2}–q^{-1/2})\frac{x^{{\mu }^{\vee }}–x^{s_i{\mu }^{\vee }}}{1–x^{–{\alpha }_i^{\vee }}}$$

${\tau }_i={\stackrel{\sim }{T}}_i\phantom{,}–\phantom{,}\frac{q^{1/2}–q^{–1/2}}{1–x^{–{\alpha }_i^{\vee }}}$ so that ${\tau }_i{x}^{{\mu }^{\vee }}=x^{s_i{\mu }^{\vee }}{\tau }_i$,

${\tau }_i^2=\frac{(q^{1/2}–q^{-1/2}{x}^{–{\alpha }_i^{\vee }})(q^{1/2}–q^{-1/2}{x}^{{\alpha }_i^{\vee }})}{(1–x^{{\alpha }_i^{\vee }})(1–x^{–{\alpha }_i^{\vee }})}$ and ${\tau }_w={\tau }_{i_1}\cdots {\tau }_{i_{\ell }}$